These motions can be produced by the linear velocities of the agents. It is known that \(\mathcal {S}_d \subseteq \mathrm {null}(\mathcal {R}_{\mathcal {G},d})\)  for any underlying graph \(\mathcal {G}\)  [1]}. For the matrix \(\bar{\mathcal {R}}_{\mathcal {G},d}\) , we can define the corresponding set
\(\bar{S}_d \mathrm {span}\left\lbrace \begin{bmatrix}1_N \otimes I_3 \\ \star \end{bmatrix}, \begin{bmatrix}\mathcal {L}^{\circlearrowright }_{\mathbb {R}^3}(\mathcal {G}) \\ \star \end{bmatrix}\right\rbrace \subseteq \mathrm {null}(\bar{\mathcal {R}}_{\mathcal {G},d}). \) 
